Categorical Statements
Categorical statements make a claim about the relationship between two classes of things — asserting that one class is (fully or partially) included in or excluded from another. There are four types, distinguished by quantity (universal vs. particular) and quality (affirmative vs. negative). Standard form: Quantifier + Subject Term + Copula + Predicate Term.
How It Appears Per Course
PHIL 252
The foundation of Unit 5’s categorical logic system. All four types are used to build categorical syllogisms and tested with Venn diagrams.
The Four Types
| Label | Form | Name | Venn Diagram Action | Example |
|---|---|---|---|---|
| A | All S are P | Universal Affirmative | Shade S-only region (empty) | All dogs are mammals |
| E | No S are P | Universal Negative | Shade overlap of S and P (empty) | No dogs are reptiles |
| I | Some S are P | Particular Affirmative | Mark “X” in overlap | Some Canadians are teachers |
| O | Some S are not P | Particular Negative | Mark “X” in S outside P | Some snakes are not venomous |
Critical notes:
- “Some” means at least one — not “most,” not “many”
- Universal statements (A, E) describe relationships only — they do not assert that S or P exists
- Particular statements (I, O) do assert existence of at least one member
Translation into Standard Form
- Rephrase terms: Use class terms (nouns), not adjectives (“silly animals” not “silly”)
- Rewrite the verb: Use “are” or “are not” + noun phrase (“Swans are swimmers” not “Swans swim”)
- Insert quantifier: Use context (definitional → All; observed patterns → Some)
- Treat proper names: “Socrates is mortal” → “All people identical to Socrates are mortal”
Always prefer affirmative predicate forms (use “non-swimmers” only when obversion requires it).
Immediate Inference Relations
| Relation | Operation | Valid For |
|---|---|---|
| Conversion | Switch S and P | E and I only |
| Contraposition | Switch S and P + replace both with complements | A and O only |
| Obversion | Change quality (aff↔neg) + replace predicate with complement | All four types |
Logical Opposition (Square of Opposition)
| Relation | Pair | Cannot both be… | Can both be… |
|---|---|---|---|
| Contradiction | A/O, E/I | True or false | — |
| Contrariety | A, E | True | False |
| Subcontrariety | I, O | False | True |
| Subalternation | A→I, E→O | — | If A true, I must be true |
Cross-Course Connections
Syllogism — categorical statements are the building blocks of syllogisms
ImmediateInference — the theory of immediate inference operates on categorical statements
Definition — the “All S are P” form captures definitional claims (e.g., “All poodles are dogs”)
ClassificationSystems — categorical statements express class membership rules
Key Points for Exam/Study
- Memorize A, E, I, O — label, form, name, Venn diagram action
- Universal statements do NOT assert existence; particular statements DO
- “Some” = at least one (this is a common trap)
- E and I are convertible; A and O are NOT
- A and O contradict each other; E and I contradict each other
- Obversion always produces an equivalent statement — applies to all four types